How can I manipulate shapes in order to have similar results as in this 3D max scene?
I guess what you need is the Mesh Deform modifier:


Create an object A and give it an Array modifier.
Create a cube B and change its mesh so that it contains your object A like a cage
In the Properties panel > Object > Display > Maximum Draw Type, choose Wire so that this cube B stays transparent in Solid mode. And in the Properties panel > Object > Cycles Settings > Ray Visibility, disable all the options so that it will be invisible in Rendered mode.
Give your object A a Mesh Deform modifier. Choose B as the Object. Click Bind.


Now, if you want to animate:


Select your object B and in the Properties panel > Data > Shape Keys, create one first shape key to save the basic shape of B, then create a second shape key and keep this second shape key selected.
In Edit mode play with the shape of the B mesh. You can see that it deforms A.
Now back in Object mode, play with the Value under the Shape Keys to change the shape of A.


And if you don't want to animate:


Just change the shape of your object B in Edit mode, it will deform object A.